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MATH7861 Discrete Mathematics
创建日期：2024-05-27 17:11:01
所属分类：数学mathematics
标签： MATH7861
Discrete Mathematics

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MATH1061/MATH7861 Discrete Mathematics Semester 1, 2023 Lecture 19 – Functions: one-one, onto, inverse

Learning Goals
Understanding what a function is, and associated concepts and definitions.
How to show functions are one-to-one, onto, bijective.
Student questions and comments:
Does the inverse function of function have the same domain and co-domain as ?
Q2 from the pre-class questions (inverse functions)
If you define the co-domain for your function differently, can you make your function become onto?
, where ଶ is not onto
ஹ଴ where ଶ is onto.
then ିଵ
Yes
Pre-work video:
Video 025 One-to-one, onto,
and inverse functions
Activity 1: Which of the following are functions?
1. where ଶ
2. ା ା where
3. ା ା where (௡ାଵ)(௡ାଷ)
ଷ
4. ା ା where (௡ାଵ)(௡ାଶ)(௡ାଷ)
଺
5. where
6. where for all with , ௠
௡
7. where
8. where
9. where ଶ
10. where
such that for each , there is a unique such that .
Challenge (optional) Activity:
Suppose and that
ଶ
for all .
Determine .
Lecture 19 – Functions: one-one, onto, inverse
ଵ
ଵ ଶ
ଶ
not one-one
not onto
range co-domain
Injective function (one-to-one function): ଵ ଶ if ଵ ଶ then ଵ ଶ
Surjective function (onto function): such that
Let be a function, .
where ଷ is surjective but not injective.
where ௫ is injective but not surjective.
Lecture 19 – Functions: one-one, onto, inverse
Poll question 1: Is a one-one function?
A. Yes
B. No
Poll question 2: Is an onto function?
A. Yes
B. No
Define a function by .
For example. and .
ଵ
ଵ ଶ
ଶ
not one-one
not onto
range co-domain
Lecture 19 – Functions: one-one, onto, inverse
Inverse Functions
Q2 from the pre-class questions
Let be a bijective function. Which one of the following is true?
a) The inverse function of does not exist.
b) For all ିଵ where is the unique element in such that
c) ିଵ is a function from to .
d) and are finite.
ଵ ଵ
ଶ
A function that is both one-one and onto is said to be a bijection.
A bijection has an inverse function ିଵ .
ଶ
ିଵ
Lecture 19 – Functions: one-one, onto, inverse
Note that the notation “ ିଵ” is used in two different ways:-
1. If is any function and is any subset of , then the
inverse image or preimage ିଵ of is defined by
ିଵ
Notice that the preimage function ିଵ is a function ିଵ
2. If is any bijective function then the inverse function of is the
function ିଵ where for all
ିଵ where is the unique such that .
Notice that the inverse function ିଵ is a function ିଵ . bijective (one-one and onto)
ିଵ
ିଵ
Lecture 19 – Functions: one-one, onto, inverse
Activity 2: For each of the following statements about the function , decide whether the
statement is true or false and give a brief reason.
(1) The inverse function ିଵ of does not exist.
(2) The preimage of the set is ିଵ .
(3) ିଵ is not defined.
(4) ିଵ .
Let be the function .
Lecture 19 – Functions: one-one, onto, inverse
ଵ
ଵ ଶଶ
not one-one
not onto
range co-domain
Injective functions (one-one functions):
To show a function is one-one, suppose that
ଵ ଶ and show that ଵ ଶ.
To show a function is not one-one, find elements ଵ ଶ
such that ଵ ଶ and ଵ ଶ.
Surjective functions (onto functions):
To show a function is onto, suppose that
and find an such that .
To show a function is not onto, find an element
such that for all .
Activity 3. Consider the function ௢ௗௗ defined by .
(a) Is an injective function? (b) Is a surjective function?
Lecture 19 – Functions: one-one, onto, inverse
Lecture 19 – Functions: one-one, onto, inverse
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